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| Mirrors > Home > MPE Home > Th. List > canthwdom | Structured version Visualization version GIF version | ||
| Description: Cantor's Theorem, stated using weak dominance (this is actually a stronger statement than canth2 9170, equivalent to canth 7385). (Contributed by Mario Carneiro, 15-May-2015.) |
| Ref | Expression |
|---|---|
| canthwdom | ⊢ ¬ 𝒫 𝐴 ≼* 𝐴 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 0elpw 5356 | . . . . 5 ⊢ ∅ ∈ 𝒫 𝐴 | |
| 2 | ne0i 4341 | . . . . 5 ⊢ (∅ ∈ 𝒫 𝐴 → 𝒫 𝐴 ≠ ∅) | |
| 3 | 1, 2 | mp1i 13 | . . . 4 ⊢ (𝒫 𝐴 ≼* 𝐴 → 𝒫 𝐴 ≠ ∅) |
| 4 | brwdomn0 9609 | . . . 4 ⊢ (𝒫 𝐴 ≠ ∅ → (𝒫 𝐴 ≼* 𝐴 ↔ ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴)) | |
| 5 | 3, 4 | syl 17 | . . 3 ⊢ (𝒫 𝐴 ≼* 𝐴 → (𝒫 𝐴 ≼* 𝐴 ↔ ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴)) |
| 6 | 5 | ibi 267 | . 2 ⊢ (𝒫 𝐴 ≼* 𝐴 → ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴) |
| 7 | relwdom 9606 | . . . . 5 ⊢ Rel ≼* | |
| 8 | 7 | brrelex2i 5742 | . . . 4 ⊢ (𝒫 𝐴 ≼* 𝐴 → 𝐴 ∈ V) |
| 9 | foeq2 6817 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝑥)) | |
| 10 | pweq 4614 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
| 11 | foeq3 6818 | . . . . . . . 8 ⊢ (𝒫 𝑥 = 𝒫 𝐴 → (𝑓:𝐴–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝐴)) | |
| 12 | 10, 11 | syl 17 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑓:𝐴–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝐴)) |
| 13 | 9, 12 | bitrd 279 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝐴)) |
| 14 | 13 | notbid 318 | . . . . 5 ⊢ (𝑥 = 𝐴 → (¬ 𝑓:𝑥–onto→𝒫 𝑥 ↔ ¬ 𝑓:𝐴–onto→𝒫 𝐴)) |
| 15 | vex 3484 | . . . . . 6 ⊢ 𝑥 ∈ V | |
| 16 | 15 | canth 7385 | . . . . 5 ⊢ ¬ 𝑓:𝑥–onto→𝒫 𝑥 |
| 17 | 14, 16 | vtoclg 3554 | . . . 4 ⊢ (𝐴 ∈ V → ¬ 𝑓:𝐴–onto→𝒫 𝐴) |
| 18 | 8, 17 | syl 17 | . . 3 ⊢ (𝒫 𝐴 ≼* 𝐴 → ¬ 𝑓:𝐴–onto→𝒫 𝐴) |
| 19 | 18 | nexdv 1936 | . 2 ⊢ (𝒫 𝐴 ≼* 𝐴 → ¬ ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴) |
| 20 | 6, 19 | pm2.65i 194 | 1 ⊢ ¬ 𝒫 𝐴 ≼* 𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ∅c0 4333 𝒫 cpw 4600 class class class wbr 5143 –onto→wfo 6559 ≼* cwdom 9604 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-wdom 9605 |
| This theorem is referenced by: pwdjudom 10255 |
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